Joe Biden has never ruled out making a third try for the presidency, and as his self-declared, end-of-summer deadline to make a decision approaches, there are fresh signals that pressure is growing on him to do so.

As the vice president endures a long and painful process of grieving after the death of his son Beau, he has received thousands of telephone calls, letters, notes and other messages offering sympathy and encouragement.

Some of those who have reached out have also urged him to run for president, aides say.

At this point, however, Biden appears to be far from any decision about whether to heed those calls.

He and his closest advisers have spoken to some of those who have suggested that he run. But there is no indication that he has taken any serious steps toward launching what would be a challenging campaign to deny Hillary Rodham Clinton the Democratic nomination.

Speculation nonetheless intensified Saturday when New York Times columnist Maureen Dowd reported that shortly before his death in May, Beau Biden encouraged his father to make another bid for the presidency. Biden’s surviving son, Hunter, also argued that he should run, Dowd wrote.

In June, the Wall Street Journal had also reported that both Biden sons wanted their father to jump into the 2016 race. Indeed, Beau Biden — a former Delaware attorney general who was himself considered a rising political star — had long favored the idea of his father running again, even when others in the family were dubious.

How Biden would fare against Clinton is questionable, given her imposing head start and the fact that both of his previous presidential campaigns, in 1988 and 2008, were wobbly, short-lived endeavors. Some advisers have already begun assisting Clinton, including Ron Klain, Biden's former vice presidential chief of staff and a longtime confidant, who is helping the former secretary of state prepare for the first Democratic debates this fall, according to senior Democrats. A Clinton campaign official confirmed Saturday that Klain has signed on with the campaign.

Those who are pushing Biden to do it argue that his supporting role in the Obama administration has added to his stature and gravitas. Moreover, he is seen as an authentic figure who connects with many voters because he is willing to show his heart — as was achingly evident after Beau’s death.

Biden will have to weigh the prospect of another campaign against what he sees as his responsibility as the family patriarch, those close to him say.

As some longtime friends have thought through his options, they have come to the conclusion that it would be preferable for Biden to concentrate fully on his final months as vice president, and potentially what role he would like to fashion for himself in the years that follow.

A campaign for president would offer a diversion from grief, they say. But he would risk some of the goodwill he has built up if he were to run badly and fall short of posing a serious challenge to Clinton.

What no one — including those around the vice president — seems to know is what his wife, Jill, thinks of the idea of another presidential campaign. The Bidens are finalizing their plans for a vacation that will gather the entire family together this month.

That, aides say, is likely to be when they have their first real discussion of whether he should run again. He does not expect to make any announcement until sometime in September.

While Clinton is the most formidable Democratic front-runner in memory, she also has vulnerabilities that worry many in her party. A number of recent polls show that her favorability is dropping amid revelations that she kept a private e-mail account while secretary of state, and the perception that she is a distant, unapproachable figure.

Meanwhile, Biden and his team had long planned a period of evaluation of his political future — one that was interrupted by trauma and tragedy.

Little by little, Biden has resumed the many responsibilities that go with his job, including the political ones. Recently, he appeared at Democratic fundraisers in Denver, Los Angeles and San Francisco, which were the first such events he had participated in since the death of his son.

“As the Biden family continues to go through this difficult time, the Vice President is focused on his family and immersed in his work,” spokeswoman Kendra Barkoff said in an e-mailed statement Saturday. “In recent weeks, the Vice President has worked on the nuclear deal with Iran, traveled across the country to highlight the Administration’s economic priorities, and more.”

Biden’s top advisers have taken account of all the important upcoming deadlines he must meet if, for instance, he decides to get on the ballot for the early nominating contests. The Democratic National Committee is also keeping Biden’s team in the loop on key decisions, such as the schedule of debates that begin this fall, and have made it clear that he would be welcome to join those onstage if he decides to run.

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